262 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS P = o NaD • In the viscous range, in which the Reynolds numbers are low, the curve has a slope of -1, which indicates that the power varies directly with vis- cosity, the square of the impeller speed and the third power of the diameter, P = I• N2D a Between these areas is a transition area where intermediate exponents apply. Once the impeller type to be used has been chosen, we have only two possible choices of the three variables, power, speed and diameter. We must choose any two of these variables to define the process job the mixer will do, then compute the third variable from the curve in Fig. 3. It is again emphasized that the curve shown in Fig. 3 does not tell us any- thing about the process job the mixer is doing. The usual method is to choose the power and impeller diameter to do the process job and then compute the impeller speed. The choice of what power and diameter to use in the process .job is the main topic of our dis- cussion here on scale-up. VISCOSITY One of the fluid properties influencing the selection of a mixer is the viscosity of the fluid. Table 1 shows the five most common classifications of fluid viscosity. Fluid viscosity is defined as the ratio of shear stress to shear rate. For example, if we apply a given shear stress to the fluid, either by a rotating impeller in a viscosimeter or a rotating mixing impeller, then the velocity pattern, or shear rate, will be determined by the viscosity of the fluid. TABLE 1.--RHEoLOGY DEFINITIONS Definition ß Effect on Viscosity,/•, of- Time Increase Shear Rate Increase Newtonian No effect No effect Non-Newtonian Pseudo-plastic No effect Decrease Dilatant No effect Increase Thixotropic Decrease No effect Rheopectic Increase No effect If at a wide variety of shear rates and shear stresses, the viscosity is constant, then we have what is called a NewtonJan fluid. If, on the other hand, at a given shear rate and shear stress, the viscosity is one value and at another rate has another value, then we have what is termed non- Newtonian behavior. Or, if the viscosity changes with time, we have non- NewtonJan behavior. If the relationship between shear rate and shear stress is as shown on Fig. 4, then we have the definitions listed. In addition, if' at a given shear

SCALE-UP CONSIDERATIONS IN COSMETIC MANUFACTURE 263 BINGHAM PLASTIC NEWTON DILATANT du SHEAR RATE • Figure 4.--Shear stress versus shear rate for various types of materials on an arithmetic plot. rate the viscosity decreases with time, then we have a thixotropic material. If the viscosity at a given shear rate increases with time, then we have rhe- opectic materials. In scale-up, these fluid properties must be carefully considered, since the difference between maximum and minimum fluid shear rates, and the ob- solute lengths of travel for the fluid are different in the two different scales. PILOT PLANTING AND SCALE-UP Mixing involves a wide variety of operations. In Table 2 are listed five major divisions into which mixing processes can be categorized. Under each fluid phase combination, we have the two basic divisions again into physical dispersion type of operations and mass transfer type of opera- tions. Physical dispersion is the physical blending or suspension of two or more phases. Mass transfer is the transfer of matter from one phase to another. Many mixing processes may involve one, two or more of these classifications or steps. For example, we may be absorbing a gas into a solid-liquid slurry with heat transfer coils to maintain temperature.